Which of the following is an even function?
f(x) = (x - 1)²
f(x) = 8x
f(x) = x² - x
f(x) = 7

Solution:

In the given problem we have to find the even function from the given functions.

A function f is even if and only if f(-x) = f(x) for all x in the domain of f, otherwise the function is odd.

f(x) = (x - 1)² = x² + 1 - 2x

⇒ Put x = -x

⇒ f(-x) = ((-x) - 1)²

⇒ f(-x) = (-x - 1)²

⇒ f(-x) = x² + 1 - 2(-x)(1)

⇒ f(-x) = x² + 1 + 2x ≠ f(x)

Therefore, f(x) = (x - 1)² is not an even function

f(x) = 8x

⇒ Put x = -x

⇒ f(x) = 8(-x)

⇒ f(x) = -8x ≠ f(x)

Therefore, f(x) = 8x is not an even function

f(x) = x² - x

⇒ Put x = -x

⇒ f(x) = (-x)² - (-x)

⇒ f(x) = x² + x ≠ f(x)

Therefore, f(x) = x² - x ≠ f(x) is not an even function

f(x) = 7

⇒ Put x = -x

⇒ f(x) = 7 = f(-x)

Therefore, f(x) = 7 is an even function

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Which of the following is an even function?
f(x) = (x - 1)², f(x) = 8x , f(x) = x² - x , f(x) = 7

Summary:

From the given functions, f(x) = 7 is the even function.